Special Cohomogeneity One Isometric Actions on Irreducible Symmetric Spaces of Types I and II

László Verhóczki

Abstract: In the present paper we study isometric actions on compact symmetric spaces for which the principal orbits are tubular hypersurfaces around totally geodesic singular orbits. We show that in these cases the symmetric space can be thought of as a compact tube the radius of which is determined by the curvature tensor. Since the constant principal curvatures of the tubular orbits can explicitly be expressed, we obtain a simple method to determine volumes of symmetric spaces by using volumes of lower dimensional ones. Finally, we discuss the classical irreducible symmetric spaces of types I and II, each of which admits such special hyperpolar actions.